The Bitcoin Equilibria Theory

Published in

Coinmonks

8 min readMar 2, 2022

My conclusion after the Bitcoin price analysis from 2015–2022.

Getting Started

When I started to interpret the Bitcoin price mathematically, I was interested in finding out if the Bitcoin price follows an ‘inner logic’, or whether it is dependent on external influences, such as the world economic situation, or similar.
After the first comparison between the value trend after the Halving of 2016 with the trend after the Halving 2020 I thought that it would be clear. The price seemed to be driven only by the intrinsic value, as also described by the stock-to-flow theory. I realized that there is a mathematical description of the price development, which could describe the long-term development as well as the medium-term growth.
But in 2021, the bitcoin price reacted differently than expected. Let’s see why.

The Equations

The long-term value trend apparently follow the general equation:
USD/BTC ≈ a SF ^ b
SF = number of existing Bitcoins / number of mined Bitcoins per year.
a = linear factor
b = exponential factor
It is justified by the stock-to-flow theory from PlanB. [1]
According to my own research, a concrete formula is 0.116 • SF ^ 3.4
The values for a and b were calculated using bitcoin prices at the time of the Halving 2016 and 2020. Why this makes sense, you will learn in this story.
My research also showed that the value trend after halving was determined by the following equation:
USD/BTC ≈ a e ^ (bt + ct³)
t = time in years since start of growth
a = start value in USD
b = starting growth factor
c = growth increase factor
Derivation in my story A Little Math and a Bitcoin Forecast [2].
Here are my concrete calculations for the bull runs in 2016/17 and 2020/21

Bitcoin hyper-exponential growth in 2016/17

Bitcoin hyper-exponential growth in 2020/21 (with way to optimistic forecast)

I also noticed that there was a lower boundary for Bitcoin value, which can be described by the following equation:
USD/BTC ≈ a e ^ bt
a = start price
b = exponential growth factor
t = time in years since start of growth
It can be derived directly from the S2F equation.
One compares the values at the Halvings with each other and calculates the exponential growth from one value to the next.
The equation for the actual Bitcoin cycle (2020 to 2024) is
USD/BTC ≈ 9100 • e ^ 0.68 t
This corresponds to a monthly increase of 5.8 % or the approximate doubling of the value in one year. If this equation is correct, the Bitcoin price at the next Halving in 2024 should be around $121000.
This path is normally best visible in the time period before the Halving.

exponential and hyper-exponential growth 2015–2018

A small note:
The behavior described is merely an observation. Unfortunately, I cannot give a reason why bitcoin behaves this way. The equations are real and can also be verified mathematically. For example, the correlation for the bull run 2020/21 with the equation found is 0.977. A coincidental match can therefore be ruled out in any case.

The Stock-To-Flow Equilibrium

We all know that PlanB’s predictions regarding his stock-to-flow theory did not materialize in 2021. According to his theory, the bitcoin price should have risen above $100000 by the end of the year 2021.
Does that mean that his theory or the equation is flawed?

Fortunately, the error is only in the interpretation of the numbers.
PlanB’s mistake was that he interpreted the values of the stock-to-flow equation as a prediction of the bitcoin price.
In reality, however, it is a description of a possible equilibrium. This means that the bitcoin price will not necessarily be reached, but rather that the values are ‘captured’ by this equilibrium when they reach the appropriate level. This can be seen very nicely in the course of the values
at the beginning of 2018. The bitcoin price was very exaggerated at that time due to the euphoria on the market, and then suddenly collapsed.
Amazingly, right at the level of the predicted stock-to-flow value.

The second mistake he made, was that he used statistical accumulations of values for his calculation instead of the very precisely known values at the time of the Halving. Thus, he included the euphoria of the market in his calculations, which generally leads to exaggerated values.

In my calculations I use a simplified method for getting the S2F-value, and assume an average mining activity. The actual value depends on the mining activity at the corresponding time. That parameter vary, and is only available on the Bitcoin blockchain. Therefore it is hard to use in a general equation. The resulting inaccuracy is negligible for my calculations.

The hyper-exponential growth Equilibrium

Any news, as well as major bitcoin buying or selling, usually only affects the price for a very short period of time, but are corrected again. Example is the Tesla investment of one billion dollars in Bitcoin, February 2021.

As the value falls back to the calculated path, it is a strong indication of an equilibrium.

My own mistake with interpreting the hyper-exponential growth, however, was that I did not initially believe that this equilibrium could simply be abandoned by external influences. As you see in the chart, however, correspondingly bad news, e.g., the crypto ban in India or in China are enough to throw the Bitcoin out of equilibrium.
Interestingly, we could subsequently see that a new equilibrium was formed, that was a hyper-exponential growth again. The price development from 20.05.2021 to 21.11.2021 clearly shows a similar development as directly after a halving.
I had to realize that such an equilibrium can also be left without a directly apparent reason, as it happened after 21.11.2021.
One can only speculate about the reasons, but the important thing in this context is that the equilibrium is completely abandoned and that there is no kind of adjustment of the price trend. There really seems to be only one way up, which is fixed from the beginning.
This way can only be broken off, an adjustment or change of the way up does not take place.
After an interruption, the upward trend starts again from the beginning, as for example after Halving. After the first growth is achieved, the logic of hyper-exponential growth takes effect again. The upward trend reinforces itself every time.

The exponential growth equilibrium

The third equilibrium is of special importance. It forms the lower limit of the Bitcoin value. If this limit is massively undershot, then a very fast upward correction takes place. The usual slow increase, as with hyper-exponential growth, does not take place here. Example of this would be the crash of 13.02.2020.

One recognizes here the immediate correction upwards. The reached lower value is not stable and is left immediately.
I think, If this would not be the case, the Bitcoin would have a serious problem, because it could leave all the stabilizing equilibria.
Undershoots are rather rare in the entire Bitcoin history, and normally takes place in the time around the halving, when the upper and lower bound gets closer.
Here, you can see very clearly that the Bitcoin price is running along a track during this period. Even larger crashes, such as on 09.11.2018 do not fall below this line and are caught by it. This is a very nice indication of its function as an equilibrium.

Summary

Overall, the following behavior of the bitcoin price can be observed:
The price usually moves between the lower equilibrium (exponential growth) and the upper equilibrium (the stock-to-flow value). Values above and below these limits are unstable, and lead to a corresponding correction.
Within these limits, the price usually moves along hyper-exponential paths, which again represent equilibria. The price can then either be caught by the stock-to-flow equilibrium, or the upward trend is interrupted and a new equilibrium is reached.
A crash, or a downward trend to the lower exponential growth is also possible, and the then existing upward trend then leads again to a hyper-exponential growth.
At the time of Halving, the lower and upper bounds coincide, so the real bitcoin price at that time is normally within a very narrow range.
The stock-to-flow equilibrium is indeed staircase-shaped and changes abruptly after halving, so that a new range of values is immediately spanned for the following cycle. in 2020 this range was between $ 9100 and $ 96000 according to my equations.

Outlook

As it currently stands, the price trend will become much more unstable in the future, especially due to the many opportunities for leveraged trading. This will cause more price drops and the value will probably fall back to the lower limit more often. The overall value trend will thus become somewhat more even, and the extreme rise after halving will thus presumably be broken off earlier. Overall, however, it looks so far as if the long-term trend, which is determined by the S2F-Theory, will be maintained.

My regular updated forecast can be found here:
www.pisu666.de
You can follow me on twitter: Pisu@Pisu6661

NOTE: This text is in no way to be considered as a financial advice. It is just my personal opinion.

References

[1] Modeling Bitcoin Value with Scarcity, by PlanB
https://medium.com/@100trillionUSD/modeling-bitcoins-value-with-scarcity-91fa0fc03e25
[2] A Little Math and a Bitcoin Forecast, by Pisu
https://medium.com/coinmonks/a-little-math-and-a-bitcoin-forecast-bcaddc17d252

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